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Bott-Borel-Weil theorem

A holomorphic action of a complex Lie group on a holomorphic vector bundle is a left holomorphic action, , which projects onto and which sends -linearly each vector space fibre onto . In this situation is conveniently said to be -equivariant. If and are equivariant bundles over and is a mapping of bundles, it is easy to see what is meant by "f is G-equivariant" and also what is meant by "E1 and E2 are equivalent" , as -equivariant vector bundles.

When , , etc. are as above, one sees that, by restriction, the given action defines a complex linear representation of the stabilizer of a point on the -vectorial fibre . The equivalence class of this representation depends only on the -equivariant holomorphism class of . If is a homogeneous -space, this correspondence between equivalence classes is bijective. This may be explained as follows: If is a complex homogeneous space and is a holomorphic complex linear representation, one considers the following equivalence relation on :

where , , . The quotient space will be denoted by , and the equivalence class of will be denoted by . The formula makes into a vector bundle of fibre type via . This fibration is naturally holomorphically -equivariant via the action and one checks that the stabilizer, , of the "neutral element" of acts (see above) on the "neutral fibre" exactly by the representation .

Below, the case will be regarded in some detail. Thus, the representation may be interpreted as a multiplicative character and will be a complex line bundle.

Background.

Let be a semi-simple complex Lie group with Lie algebra (cf. also Lie group, semi-simple; Lie algebra), a Cartan subalgebra of , a system of positive roots (cf. Root system), the corresponding system of opposite roots (termed negative), and the set of all roots. Let be the root space associated to . Then is a nilpotent Lie subalgebra and one defines the maximal solvable subalgebra (the Borel subalgebra) by . This is the Lie algebra of a closed complex Lie subgroup such that is compact. Finally, .

Note that there is a subspace of that is, in the vector spaces sense, a real form (that is, and is the Lie subalgebra of a compact connected group). It follows that the restriction to of the Killing form of the complex algebra, denoted by , is a real scalar product. From this one deduces an isomorphism and thus a scalar product on . Notice that the evaluation of the weights of representations (and also of the roots) on are real numbers. Recall that the closed Weyl chamber is the set of for which for all . The Weyl group acts on , with as "fundamental domain" . It is worth noting that while the transformation is not necessarily in the Weyl group, the opposite is the transformation of by an element of the Weyl group (in fact, by the longest element). Now consider an irreducible representation . The theory of H. Weyl classically characterizes such a representation by its dominant weight (cf. also Representation of a Lie algebra). Contrary to tradition, it is perhaps wiser to characterize a representation by its dominated weight. This is the unique weight of the representation such that the other weights of may be obtained from by the addition of an -linear combination of positive roots. In general, the dominated weight of a representation is not the opposite of the dominant weight of , but the opposite of the dominant weight of the contragredient representation . This dominated weight is always in the opposite of the Weyl chamber.

Bott–Borel–Weil theorem.

In the above context, consider the hyperplane that is the sum of all the proper spaces associated to the weights different from the dominated weight of the representation . By the definition of dominated weight, one sees that . Now consider the holomorphically trivial bundle , and make it equivariant by the action . This -equivariant bundle is exactly , which leads to the equivariant exact sequence of holomorphic bundles:

In fact, the weight extends to a character , which can be integrated to give a character . One easily sees that and that the natural action of on is exactly the representation .

In this context, the Borel–Weil theorem states:

a) The arrow is a -equivariant isomorphism;

b) for . These results are not unexpected (in case b), at least for those who are familiar with the idea of a sufficiently ample line bundle). This is not at all the case for the generalization to representations of in when the line bundle is given by a representation such that the restriction to of its derivative is not the dominated weight of a holomorphic representation of . Indeed, this generalization is the very unexpected Bott–Borel–Weil theorem: Let , and be as above, and let also be the Weyl group relative to the Cartan algebra and . Then:

i) If, for all , the quantity is never the dominated weight of a representation, then all the cohomology groups are zero.

ii) If there exists an element , hence unique with this property, such that is the dominated weight of a representation , then:

A) For (the length of ), the cohomology group is zero.

B) For , the natural representation of on the cohomology group is exactly the representation .

The proof is essentially a very beautiful application of the relative cohomology of Lie algebras, initiated by C. Chevalley and S. Eilenberg.